A gas, initially at T=500 K, undergoes a Joule-Thompson (constant enthalpy) expansion from P=10 atm to P=1 atm.
A) Prove that (dT/dP)H is equal to: (1/Cp)[T(dV/dT)P – V]
(You may assume this expression is valid for parts B and C, even if you are unable to prove it.)
B) Find an expression for (dV/dT)P for this gas, in terms of P, T and known constants, assuming it follows the equation of state: V = (RT/P) + aT2P2
For full credit your answer should be an algebraic equation—all derivatives should be evaluated.
C) Assuming the heat capacity of the gas is constant at Cp=5R, and that ‘a’ is constant at
2x10-7 L mol-1K-2atm-2, find the final temperature of the gas.
Where (?P/?P)_T=1 , ((??H)/?P)_?S=?V , and ((??H)/(??S))_P=T from the fundamental property relationship for change in enthalpy:
Where ((??S)/?P)_T=-((??V)/?T)_P from Maxwell’s relationships:
Substitute ((??H)/?P)_T back into (?T/?P)_?H expression:
Write (?T/?P)_?H in terms of P and evaluate to find the final temperature of the gas:
This can be solved as a quadratic equation, or you can recognize that the first term on the right hand sides proves to be insignificant for realistic values of T:
T_2=459.4 K
(The other solution to the quadratic equation is T2=6.154x107 K.)
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